Rút gọn biểu thức:
I = $\frac{(\sqrt[]{x}-\sqrt[]{y})^2+ \frac{2x^2}{\sqrt{x} }+ y\sqrt{y} }{x\sqrt{x}+y\sqrt{y}}+\frac{3\sqrt{xy}-3y}{x-y}$
( x>0 , y>0, x $\neq$ y)
Mình cảm ơn.
Rút gọn biểu thức:
I = $\frac{(\sqrt[]{x}-\sqrt[]{y})^2+ \frac{2x^2}{\sqrt{x} }+ y\sqrt{y} }{x\sqrt{x}+y\sqrt{y}}+\frac{3\sqrt{xy}-3y}{x-y}$
( x>0 , y>0, x $\neq$ y)
Mình cảm ơn.
Đáp án:
\(\dfrac{{x – 2\sqrt {xy} + y + 2x\sqrt x + 4y\sqrt y + 3x\sqrt y – 3y\sqrt x }}{{x\sqrt x + y\sqrt y }}\)
Giải thích các bước giải:
\(\begin{array}{l}
I = \dfrac{{{{\left( {\sqrt x – \sqrt y } \right)}^2} + 2x\sqrt x + y\sqrt y }}{{x\sqrt x + y\sqrt y }} + \dfrac{{3\sqrt y \left( {\sqrt x – \sqrt y } \right)}}{{\left( {\sqrt x – \sqrt y } \right)\left( {\sqrt x + \sqrt y } \right)}}\\
= \dfrac{{x – 2\sqrt {xy} + y + 2x\sqrt x + y\sqrt y }}{{\left( {\sqrt x + \sqrt y } \right)\left( {x – \sqrt {xy} + y} \right)}} + \dfrac{{3\sqrt y }}{{\sqrt x + \sqrt y }}\\
= \dfrac{{x – 2\sqrt {xy} + y + 2x\sqrt x + y\sqrt y + 3\sqrt y \left( {x – \sqrt {xy} + y} \right)}}{{\left( {\sqrt x + \sqrt y } \right)\left( {x – \sqrt {xy} + y} \right)}}\\
= \dfrac{{x – 2\sqrt {xy} + y + 2x\sqrt x + y\sqrt y + 3x\sqrt y – 3y\sqrt x + 3y\sqrt y }}{{\left( {\sqrt x + \sqrt y } \right)\left( {x – \sqrt {xy} + y} \right)}}\\
= \dfrac{{x – 2\sqrt {xy} + y + 2x\sqrt x + 4y\sqrt y + 3x\sqrt y – 3y\sqrt x }}{{x\sqrt x + y\sqrt y }}
\end{array}\)